Select all the expressions that are equivalent to 8^5/4^5. Multi-select Choices: (A)1/2^5 (B)1/2^-5 (C)2 (D)2^5

Simplify Exponents: Simplify the given expression using the properties of exponents. $ 8^5 / 4^5 = (2^3)^5 / (2^2)^5 $ Using the power of a power rule, $ (a^m)^n = a^{m*n} $, we get: $ (2^3)^5 = 2^{3*5} = 2^{15} $ $ (2^2)^5 = 2^{2*5} = 2^{10} $ So, $ 8^5 / 4^5 = 2^{15} / 2^{10} $Power of Power Rule: Continue simplifying using the quotient of powers rule. Using the rule $ a^m / a^n = a^{m-n} $, we get: $ 2^{15} / 2^{10} = 2^{15-10} = 2^5 $Quotient of Powers Rule: Compare the simplified expression to the given choices. $ 2^5 $ is equivalent to: (A) $ 1/2^5 $ - This is $ 2^{-5} $, not $ 2^5 $. (B) $ 1/2^{-5} $ - This is $ 2^5 $, because $ 1/2^{-5} = 2^5 $. (C) $ 2 $ - This is $ 2^1 $, not $ 2^5 $. (D) $ 2^5 $ - This is exactly $ 2^5 $.

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Algebra 2

Simplify rational expressions

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Q. Select all the expressions that are equivalent to

\frac{8^5}{4^5}

\newline

Multi-select Choices:

\newline

(A)

\frac{1}{2^5}

\newline

(B)

\frac{1}{2^{-5}}

\newline

(C)

2

\newline

(D)

2^5

Simplify Exponents: Simplify the given expression using the properties of exponents. $\newline$ $8^5 / 4^5 = (2^3)^5 / (2^2)^5$ $\newline$ Using the power of a power rule, $(a^m)^n = a^{m*n}$ , we get: $\newline$ $(2^3)^5 = 2^{3*5} = 2^{15}$ $\newline$ $(2^2)^5 = 2^{2*5} = 2^{10}$ $\newline$ So, $8^5 / 4^5 = 2^{15} / 2^{10}$
Power of Power Rule: Continue simplifying using the quotient of powers rule. $\newline$ Using the rule $a^m / a^n = a^{m-n}$ , we get: $\newline$ $2^{15} / 2^{10} = 2^{15-10} = 2^5$
Quotient of Powers Rule: Compare the simplified expression to the given choices. $\newline$ $2^5$ is equivalent to: $\newline$ (A) $1/2^5$ - This is $2^{-5}$ , not $2^5$ . $\newline$ (B) $1/2^{-5}$ - This is $2^5$ , because $1/2^{-5} = 2^5$ . $\newline$ (C) $2$ - This is $2^1$ , not $2^5$ . $\newline$ (D) $2^5$ - This is exactly $2^5$ .